by Michael Tao - Dynamic Graphics Projectmtao/files/publications/msc.pdf · to the existing technologies in graphics for simulating granular materials, we choose a continuum based

Coupled Simulation of Fluids and Granular Materials

by

Michael Tao

A thesis submitted in conformity with the requirementsfor the degree of Master of Science

Graduate Department of Computer ScienceUniversity of Toronto

c© Copyright 2014 by Michael Tao

Abstract

Coupled Simulation of Fluids and Granular Materials

Michael Tao

Master of Science

Graduate Department of Computer Science

University of Toronto

2014

This thesis presents a novel method for the simulation of mixed fluids and granular materials. In contrast

to the existing technologies in graphics for simulating granular materials, we choose a continuum based

approach in order to capture interesting phenomena with a large number of particles. By extending

pre-existing models for simulating both fluids and granular materials as continuua, we generate novel

phenomonology in our simulations of both fluids and granular materials, including their interactions

with each another.

Contents

1 Introduction 5

1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Preliminaries 7

2.1 Constitutive Equations for Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2.1 Lagrangian and Eulerian Perspectives . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3.1 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3.2 Free Surface Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Darcy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.5 Constitutive Equations for Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . 12

2.1.6 Continuum Behavior of Sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.6.1 Unilateral Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.6.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Relevant Work 16

3.1 Fluid Simulation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Eulerian Grid Fluid Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.2 Particle Fluid Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.3 Semi-Lagrangian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Granular Simulation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Particle Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.2 Continuum Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Porous Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Overview of Simulation Methods 21

4.1 Operator Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Cubical Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Levelset Based Hodge Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.0.1 Length of edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2

CONTENTS 3

4.2.0.2 General Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.0.3 Dealing with Degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.1 Pressure Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.2 Free Surface Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Darcy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.1 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.2 Porosity Based Hodge Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.3 Capillary Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.3.1 Maintaining Particle Density . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.4 Cohesive Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4.1 Isotropic Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4.2 Deviatoric Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4.3 Using Levelset Hodge Star for UIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.5 Wet Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.5.1 Ordering Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.5.2 Advecting quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Technical Details 31

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Modified MPRGP Quadratic Programming Solver . . . . . . . . . . . . . . . . . . . . . . 31

5.2.1 Convergence Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.2 Projection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.4 Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2.5 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Results 36

6.1 Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.2 Porous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.2.1 2D Rendering Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.3 Combined Fluid and Granular Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7 Concluding Remarks 41

7.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.1.1 Stability Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A Calculus of Differential Forms 43

A.1 Ordinary Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.2 Tangent Bundles on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.3 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A.4 Differentiating Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

A.5 The Hodge Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

CONTENTS 4

A.6 Closed and Exact Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A.6.1 Hodge-Helmholtz Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

A.7 Integrating Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

A.7.1 Cube Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

A.7.2 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A.7.2.1 Laplace-de Rham Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A.7.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

A.8 Discrete Calculus on Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

A.8.1 Exterior Derivative and Boundary Operators . . . . . . . . . . . . . . . . . . . . . 50

A.8.2 Hodge Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A.9 Pressure Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

B Discrete Differential Forms on Cubical Complexes 51

B.1 Level-set based Hodge Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

B.1.1 Connection to Staggered Grid Methods . . . . . . . . . . . . . . . . . . . . . . . . 51

B.1.2 Boundary Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

B.1.3 Hodge Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

B.1.3.1 Hodge Star on Uniform Grids . . . . . . . . . . . . . . . . . . . . . . . . 52

B.1.3.2 Differential and Codifferential Operator . . . . . . . . . . . . . . . . . . . 53

B.1.3.3 Constructing the Laplace-deRham . . . . . . . . . . . . . . . . . . . . . . 53

C Numerical Optimization 54

C.0.4 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

C.0.5 Linear Complementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

C.0.6 LCP and Karush-Kuhn-Tucker Conditions . . . . . . . . . . . . . . . . . . . . . . . 56

C.0.7 Complimentary and Minimization Duality . . . . . . . . . . . . . . . . . . . . . . . 56

Chapter 1

Introduction

Uncountably many films have scenes that contain beaches, but there is rarely, if ever, a proper simulation

performed on the large body of sand. That is because simulating the interactions of sand particles is

an intractable problem: there are simply too many particles that can interact with one another. Even

if we could make the simulation of sand tractable, we would have only shown how to simulate a desert.

Without being able to simulate water, and its interactions with sand, we would not have a method for

generating beaches.

Sand is a granular material; this family also includes materials like rice and cornstarch, which are

ubiquitous materials in human the world. By taking into account the prevalence of water and other fluids

on our planet, the combined dynamics of granular materials and fluids are some of the most common

phenomena on the planet. In fact there are many recreational human applications of this multi-material

interaction such as sand castles and cooking, as well as industrial applications in the design of crucial

infrastructures such as water filtration systems and bridges.

Although the combination of granular materials and fluids has such great presence in our lives, there

are only a few methods available for simulating them at a scale of more than thousands of particles,

which wouldn’t be able to even fill a pail of sand in a sandbox. The reason for this lack is multi-fold,

but mostly it derives from a variety of representational issues.

The natural way one might think of granular materials is as a collection of its constituent granules.

That is, a granular material is a collection of small, relatively rigid, particles that interact with each

other through rigid body collisions and conservation of momentum. Though this solution is direct and

obvious, it is not easily scalable because the forces generated by any single particle can perturb the

whole system, which implies that we would need a global solution for the dynamics of each simulation

timestep. Indeed, advecting a body of granules is a global problem, where the force from one particle

can easily affect fairly distant particles as one can readily see in an apparatus such as Newton’s cradle.

Furthermore, the granule-based approach is not amenable to some standard representations of fluids.

If one were to choose a particle-based method the fluid particles would have to be significantly smaller

than the size of the granules in order for the fluid particles to be able to advect through a mass of

granular particles. With the choice of grid or mesh based methods, there are several issues that arise

due to the fundamental differences between the sort of quantities stored. Communication between the

granules and the fluid stored on the grid would lead to a fair amount of averaging and interpolation

between the grid and the particles.

5

Chapter 1. Introduction 6

Now if we instead consider the option of treating granules as a continuum, we are quickly confronted

withtwo fundamental issues. The first is that much of the interesting behavior from granular materials

comes from the behavior of individual particles, which can quickly disappear as “interesting” simulation

scales would require particles to be at such fine scales so as to would make them invisible. The second,

far more glaring issue is that there is no general consensus of a single model for granular materials as a

continuum. There are too many variables available such as granule distribution and granule geometry

to take into account that can change the behavior of the granular material as a whole and the addition

of fluids doesn’t help with the situation.

In this thesis we discuss how one can ameliorate some of the issues that one encounters through a

hybrid simulation method that advects individual granules, and allows for granule-granule interactions,

but also takes into account global scale effects through a grid to maintain global effects, which include

the addition of fluid matter into the bodies of granules.

1.1 Outline

This thesis is organized as follows:

• Chapter 2 reviews some rheological preliminaries, with Appendix A to provide some further math-

ematical preliminaries.

• Chapter 3 provides comparison with previous work in the field.

• Chapter 4 overviews of the method in fairly general terms.

• Chapter 5 describes the method in its technical details.

• Chapter 6 finishes the thesis with some results and final remarks.

Chapter 2

Preliminaries

In order to develop an intuition for the nature of granular materials, fluids, their mixtures, and how to

simulate them we will first explore some necessary physical and mathematical foundations. This section

will provide an overview of the intuition behind the various data structures and equations that are used

later. We choose to use the language of exterior calculus and differential forms in our mathematical

descriptions. This choice allows us to use a simple and elegant family of linear differential operators that

allow for a significant amount of deformation in the space. For completeness we have included a brief

overview of the those topics in Appendix A.

The two categories of material we will discuss are what are frequently called Newtonian and a dilatant

Non-Newtonian fluid. Though we will go into their definitions in further detail in the following sections,

Newtonian fluids are what one normally thinks of as a fluid such as water and oil. Other types of

fluid are called Non-Newtonian fluids, where dilatant (or shear thickening) fluids are a specific type of

Non-Newtonian fluid that act more like a solid when a moderate amount of pressure is applied to it. In

our case we are looking at sand, but dilatancy is a common feature in different types of mixtures such

as blood, ketchup, and oobleck

We will first tread through the more commonly known constitutive equations used for Newtonian

fluid simulation, then expand on those equations to generate constitutive equations for more general

flows.

2.1 Constitutive Equations for Newtonian Fluids

Deriving a model for the dynamics of fluids begins by trying to tracking infinitesimal parcels of fluid

material while guaranteeing that basic material laws are followed. In particular, all materials must

satisfy the conservation laws of mass and balance of momentum. With just those two conservation laws,

viscosity, and incompressibility we can obtain the standard equations used for modeling Newtonian fluids

– the Navier-Stokes Equations.

7

Chapter 2. Preliminaries 8

2.1.1 Conservation of Mass

The amount of mass of fluid in a particular volume Ω can be identified by the integral of density ρ

defined over Ω through the integral

masspΩq “

ż

Ω

ρdV

Conservation of mass is precisely stated by looking at the time derivative of mass:

0 “B

Bt

ż

Ω

ρdV “ ´

ż

BΩ

ρu ¨ n dA.

where u is the velocity of mass, n is a the unit outwards normal vector, dV is the volume. Because

u ¨ n represents the “amount of velocity going outside on the boundary of Ω,” the right hand side can

be interpreted as the “amount of density moving outside on the boundary of Ω.”

By using the divergence theorem and moving everything to the left hand side we obtain

ż

Ω

Bρ

Bt`∇ ¨ pρuqdV “ 0

Because this equation applies to any domain X Ă Ω, we can write this integral equtation as a differential

equtation:Bρ

Bt`∇ ¨ pρuq “ 0

which is called the continuity equation. Quite simply, the change in density at a point is equal to the

amount advected through that point, so no material will spontaneously appear.

2.1.2 Conservation of Momentum

Fluids follow the Newton’s famous equation for balancing momentum: F “ ma. When we write it by

tracing the forces applied to an infinitessimal parcel of material, treated as a particle:

F “ ρDu

Dt

where DDt is the material derivative, which represents the change in a function for a specific particle.

This special derivative takes into account the fact that the derivative we evaluate at a particular point

in space/time does not correspond to the derivative of a value on a particular location. For example, if

one were to place a depth sensor on a boat and allow it to sail into an ocean, the sensor would probably

detect changes in depth, resulting in a nonzero derivative for the depths recorded by the sensor. The

material derivative would take into account the velocity of the boat, causing the derivative to be zero.

This virtual particle is best explained through an exposition of the Eulerian and Lagrangian perspec-

tives, as we now discuss.

2.1.2.1 Lagrangian and Eulerian Perspectives

As a fluid is advected in a domain there are two natural ways to record quantities of those fluids as they

move: the Lagrangian perspective and the Eulerian perspective. In the Lagrangian viewpoint the fluid

is represented as a collection of constituent particles. Dynamics are performed by moving the positions

of particles in space through time, similar to the boat in the previous example. The Eulerian viewpoint


maintains measurements of quantities at fixed positions in time, similar to the ground in the previous

example.

The material derivative is simply derived by utilizing the chain rule. Taking the derivative of a

scalar quantity q being advected along a velocity field u we define the material derivative to be the time

derivative of the quantity at a particular point and time:

Dq

Dt“

d

dtqpt, xq “

Bq

Bt`ÿ

i

Bq

Bxi

dxidt“Bq

Bt`∇q ¨ u.

The material derivative is split into two terms: the change of a quantity at a point, BqBt , and the offset

caused by the quantity being advected by u, ∇q ü. This second term can be seen as correction required

to translate between the Eulerian and Lagrangian perspectives.

Assuming incompressibility is the same as assuming DρDt “ 0, we obtain, through subtraction with

the continuity equation and the chain rule, that

0 “Dρ

Dt´Bρ

Bt´∇ ¨ pρuq

0 “Bρ

Bt`∇ρ ¨ u´ Bρ

Bt´∇ ¨ pρuq

0 “ ρ∇ ¨ u

0 “ ∇ ¨ u.

Such a velocity field is said to be divergence-free. Requiring the velocity field to be divergence-free is

often called the incompressibility condition.

2.1.3 Forces

In the Navier-Stokes equations there are three types of forces that direct the motion of fluid: pressure,

viscosity, and external forces. The force generated by pressure is the force pushing a fluid from high

pressure regions to low pressure regions and the force from viscosity is the fluid equivalent of a frictional

force from within the fluid. External forces will generally mean the sum of body forces including gravity

and boundary forces such as interactions with rigid bodies.

Computationally, viscosity and pressure are typically separated. This comes from Chorin’s splitting

method[Cho67], where the computation required to maintain incompressibility and to generate viscous

forces are separated. Splitting is crucial to maintain incompressibility, and also allows for the trivial

removal of viscosity in situations when simulating fluids that have a low viscosity. These fluids are

called inviscid fluids, and are very common in graphics because of water’s low viscosity and the cost of

numerically solving for viscosity would suggest taht additional computation be used only when necessary.

2.1.3.1 Pressure

In an incompressible fluid, pressure can be seen as the force that maintains the incompressibility con-

straint, so pressure can be computed as the force returning ∇ ¨ u to zero. After applying the various

other forces present in a system, the velocity field for a fluid tends not to be incompressible anymore. A

simple way to maintain incompressibility is to find new velocity fields to substitute the invalid velocity


fields, which can be done through a projection in the L2 sense from the invalid velocity field back into

the space of gradient-free vector fields.

We find that the best derivation of this is done with differential forms. In the language of differential

forms the incompressibility condition can be written as du5 “ 0. When it is not ambiguous, we will omit

the ¨5 from u because the contexts for when u is treated as a 1-form and vector field are unambiguous.

From the perspective of differential forms the incompressibility condition is the same as u being co-closed.

This is easily seen by considering the Hodge-Helmholtz decomposition A.6.1 that any any differential

k-form can be written in terms of a k´ 1-form, k` 1-form and a k-harmonic form. If we set the 0-form

in our case to be labeled as p for pressure, we see that

δu “ δdp` δδω ` γ

δu “ δdp

Therefore u ´ dp “ δω ` γ represents a divergence-free vector field, which is therefore incompressible.

This definition of pressure treats can be treated as a projection operator of a velocity field u to the

L2-nearest incompressible. The force necessary to set ∇ ¨ u is therefore ´∇p.For a more traditional vector-calculus perspective, one can check our discussion of differential forms A.4

to see that δu5 “ ∇ ¨ u as well as that pdpq7“ ∇p to derive

δu “ δdp

∇ ¨ u “ ∇ ¨∇p,

which is the standard pressure projection problem.

2.1.3.2 Free Surface Conditions

One crucial component for interesting fluid simulations is to have a visible fluid boundary. In order to

produce such a simulation there are some minor changes required in the pressure solving step to take

into account the incompressibility of the system while handling the boundary. One standard solution is

to set the pressure of the air to be 0, which we will utilize and go into more detail in the Overview 4.

2.1.3.3 Viscosity

Viscosity can be seen as a sort of friction between infinitesimal particles. The force effectively reduces

the amount of shearing that the fluid can do, which homogenizes fluid flow. That is, it pushes the

velocity field of a fluid to be closer the local average velocity. It is essentially a diffusion of momentum

in a fluid. The force from viscosity does not satisfy an intrinsic constraint like incompressibility, but is

a force returning to a body to counteract shearing in the material.

Viscosity is most readily defined on the boundary as a tensor σ such that for a point on the boundary,

with normal n, the viscosity force is σn. Unlike pressure, where we used a scalar value p, σ has to be a

tensor to take into account the fact that the force of viscosity will not be parallel to n.

The local changes in u are represented by ∇u, so viscosity forces must be representable as a function

of ∇u. As a first order approximation we will assume that σ is linear with respect to ∇u. Because of

constraints such as the symmetry and rigid-body invariance of σ, not all of ∇u is relevant, and it is


possible to remove parts of the operator that have no bearing to the computation of viscosity. First,

consider the decomposition

∇u “ ∇u`∇ut

2`∇u´∇ut

2“ D ` S.

Because rigid rotations should not have an affect on the relative velocities, we can ignore S and only

pay attention to D.

Because D is symmetric, that is, it is invariant on the order of its vector arguments, it can be

diagonalized. In order to be symmetric on each of the axes, the operator from D to σ can only be a

scaling of D and an added diagonal term. The only possible linear term to add to the diagonal is a

scaling of ∇ ¨ u in order to add the different diui terms symmetrically. We therefore have a form like

σ “ λ∇ ¨ uI` 2µD

Because we are looking at an incompressible fluid, ∇ ¨ u “ 0, the leftover terms of viscosity will be

σ1 “ µD.

By applying the divergence theorem in the same way utilized for pressure, we see obtain a body term of

µ∇ ¨D “ µ∇ ¨∇u`∇ ¨ p∇utq

2“ µ∇ ¨∇u.

The viscous Navier-Stokes equations therefore have the following form:

ρDu

dt“ ∇ ¨ τ ´∇p` g

∇ ¨ u “ 0

τ “ µp∇u` p∇uqT q

where g are all external forces, usually including gravity, and µ is the viscosity coefficient, which is a

property of the fluid being simulated.

2.1.4 Darcy Flow

Fluids in a porous medium are by definition surrounded by solid matter so the velocity of a wet porous

mass should be approximately the same as the velocity of the particles and porous material. In this sort

of flow, which is commonly called Darcy flow, we can therefore assume that the momentum is eliminated

by the solid medium. When the porous medium is not moving in space, one can represent the removal

of momentum as the Eulerian and Lagrangian perspectives being identical. That is, we will assume that

Dq

dt“Bq

dt

in the context of Darcy flow.

We also add a force of capillary action, which is defined by a surface potential that defines the affinity

of the surface of granules with whatever fluids are interacting with them. Capillary action is what allows

for water to diffuse against the flow of gravity, through porous materials like sand, sponges, and paper.


Though capillary action is a surface-to-surface interaction, at a continuum level it is treatable as a

volumetric quantity through homogenization, utilizing some assumptions on the porous body’s internal

geometry [Bea72]. For instance one can assume that the interior of a sponge is a series of spheres or

cylinders to generate different models for the volume to surface area ratio.

With those two changes, we obtain the following equations for incompressible flow:

Bu

Bt“ ´

κ

µ∇p` κ

µρg

∇ ¨ u “ 0

g “ ´∇Φ` g

where κ is permeability, µ is viscosity, and ´∇Φ is capillary potential, and g is usually gravity. This

same procedure has been seen applied to heat transfer in [?]. We will suppress the above fraction with

the variable κ “ κµ .

2.1.5 Constitutive Equations for Non-Newtonian Fluids

The rhelogical behavior of sand is quite different from water. At the continuum level this difference is

often mentioned as the differences between a Newtonian and a non-Newtonian fluid, with water being the

former and sand being the latter. What determines whether a fluid is Newtonian or not is its response

to shearing.

In the case of a Newtonian fluid stress is a linear response to shearing. The behaviors described by

this model are the standard ones one imagines for a fluid like water: the resistance one meets when

pushing against the fluid is proportional to the relative speed of the fluid itself.

Any fluid with a nonlinear stress/strain relationship is considered non-Newtonian. In the non-

Newtonian case, the viscous stresses can depend on a variety of other factors such as pressure, tem-

perature, or electromagnetic effects. The behaviors are much more varied than those of Newtonian

fluids. With shear thickening fluids like cornstarch suspensions, beyond some amount of pressure on the

fluid the force required to move through the material becomes substantially more difficult. It is even

possible to run on a pool filled with cornstarch suspended in water.

On the other hand, shear thinning fluids like ketchup seem somewhat rigid at rest but will become

fluildized when pressure is applied. This particular behavior is why many modern condiment containers

have caps on the bottom: gravity applies insufficient pressure for the ketchup to fluidize so it does not

escape the container, but once some pressure is applied the ketchup will fluidize and squirt out.

2.1.6 Continuum Behavior of Sand

The rheological behavior of granular materials at a continuum scale is that of a time independent

viscous non-Newtonian fluid; in particular they are shear thickening. The structure of the relatively

rigid individual particles aggregates frictional forces so that, past a specific amount of pressure, a given

amount of shear stress will not generate further shearing. Because of its time independence, the flow of

a granular material can be locally defined as an energy minimization based on its stress tensor σ which

we separate into the isotropic mean stress term (pressure) p and the traceless deviatoric stress term

(frictional) s:

σ “ ´pI ` s.


This is similar to the decomposition that we saw before for incompressible fluids, but here we do not

force the frictional component to be linear. Sand can be treated as being incompressible, so pressure

should be the same linear equations as before, while the s is not due to the existence of two different

modes of friction that exist: static and kinetic.

The boundary between the two frictional regimes can be expressed through a yield criterion. The

criterion demarks a boundary constraint on the stress tensor before the granular material acts in a rigid

regime. Intuitively what we want to represent is that a sufficient amount of shearing stress must occur

before any motion happens, and that the yielding point increases monotonically with pressure. Usually

the yielding point is linear to pressure. Here we use the Drucker-Prager yield condition, a continuous

variant of the Mohr-Coulomb yield condition [ZB05]:

s ď?

3αp` c

for some norm ¨ (we will use the Frobenius or L2 norm) and α, c being material parameters. The

term c corresponds to cohesion, which I will assume is 0 to simulate dry granular materials. α has a

geometric interpretation: if the steepest angle that granules can be piled, the angle of repose, is θ, then

α “a

23 sin θ.

The method we use alternates between solving for a velocity that satisfies pressure constraints and

solving for frictional constraints using a variational principle.

2.1.6.1 Unilateral Incompressibility

In the above framework the static representation of granular material is stored precisely as that of a

viscous fluid. The only difference represented is the yield condition that appears in the resolution of

the stress. However, by treating granular materials like a viscous fluid, some crucial information on the

incompressibility of individual grains is lost. Traditional fluid incompressibility does not assume any

atomic set of spherical particles, and each parcel of fluid is allowed to deform into arbitrary shapes and

aspect ratios. Granular materials, however, consist of real physical particles that cannot deform and

so a standard pressure solve is not sufficient. Resolving particle collisions is a global problem because

forces are propagated at a much higher speed than our intended timesteps, so some form of global

incompressibility solution is required. Narain et al. [NGCL09] introduce the Unilateral Incompressibility

Constraint (UIC) in order to roughly solve this problem at a global scale. The intent is to use continuum

methods to find a rough but adequate global solution using a cheap local solver to find a reasonable

solution.

Enforcing the Unilateral Incompressibility Condition, which represents the maximal density obtained

when packing spheres, at global scale seems to be a decent candidate for the approximate solution we

seek. The condition explicitly sets a maximal density

ρmax “ 2αp?

3d2minq

for a parameter α representing imperfections in packing and dmin representing the minimal distance

allowed between objects (the radius of a particle of sand). When the density of a grid cell is below the

packing density, the system assumes that there is no need to interrupt the advection of particles in that

particular grid cell. However, if the particle density increases to beyond the maximal density, the system

generates a pressure force to guarantee that by the next timestep the density is lowered to a valid level.


Within a homogeneous medium and given a density constraint, the mass of material in a finite volume

can be identified with a volume fraction φ, and the momentum of that mass can be identified with φv.

In [NGCL09] the advection of density through pressure is given by

φn`1 “ φn`1|p“0 ´∆t2

ρmax∇2p

where φn`1|p“0 is the density at the next time step ignoring pressure. The complimentary constraint

pp1´φn`1q “ 0, implies that a nonzero pressure will exist if and only if the system is completely packed

in that cell. We can therefore formulate this as a linear complementary problem (LCP):

A1p` b1 ě 0 (2.1)

p ě 0 (2.2)

pT pA1p` b1q “ 0 (2.3)

where

A1 “∆t2

ρmaxD1

TD1 (2.4)

b1 “ 1´ φn`1|p“0 (2.5)

given that D1 is the finite difference gradient operator on the density. For more information on LCP

problems refer to Appendix C.

This method is sufficient for guaranteeing that particles do not usually get too close but it is not an

exact solution to completely avoiding self intersection. Therefore, after each UIC solve the simulation

usually pushes each particle around in a small random walk, which works because the particles are

already mostly separated out and only need to be jittered slightly.

2.1.6.2 Friction

Frictional stress is computed by minimizing the kinetic energy in a system:

E “1

2

ż

ρnv2dV.

For computational purposes a reweighting is performed for w “ ρnρmax

E “1

2

ż

wρnv2dV.

This modification introduces some error but improves the conditioning of the problem by reducing the

variation of coefficients used by low-density portions of the discretized system. The constraints of the

Druker-Prager yield criterion are approximated with a series of planar constraints by bounding the

coefficients of the tensor by smax “ αp [NGCL09]. Minimizing this energy functional is a quadratic so

we obtain the following quadratic system:


sTA2s` bT2 s ě 0 (2.6)

smax ě s ě ´smax (2.7)

where

A2 “∆t2

ρmaxDT

2 D2 (2.8)

b2 “∆t

ρmaxD2

T ρnv|s“0 (2.9)

given that s is a vector of the stress tensor coefficients and D2 is the finite difference gradient operator

on the stress tensor. This system is easily transformed into a linear complementary problem by applying

the Karush-Kuhn-Tucker conditions, which are an extension of λ-multipliers that takes into account

boundary conditions.

2.2 Summary

In this chapter we first derived the necessary equations for describing the dynamics we will discuss in

later sections. We first derived standard fluid flow equations to provide some basic insight on fluid

simulation and how the dynamics are conventionally separated. From there, we provided alternative

models for pressure and for viscosity to present a model for simulating granular materials, including

some numerical details. For details on the mathematics we will use or for further information on LCP

problems please refer to the appendices (Appendix A and Appendix C respectively).

Chapter 3

Relevant Work

The rheology of non-Newtonian fluids has only recently appeared in computer graphics literature [GBO04],

with wet granular materials only appearing as recently as 2008 [RSKN08]. Non-Newtonian fluids in the

general sciences are a well known topic, though the precise mechanisms behind how they operate are still

open for discussion. There have been some results in non-physical simulation of sand and mud such as

the work of [SOH99], but we will primarily pursue the physical aspects of simulation. Cohesive granular

materials and porous flow have a strong history in engineering, in areas such as soil mechanics [Har25]

but we will focus our scope on simulations of visual phenomenology and thus focus on the computer

graphics literature. We primarily restrict our discussion to publications from the domain of graphics.

3.1 Fluid Simulation Techniques

The first attempts to fully simulate the Navier-Stokes equations in graphics literature was by Foster and

Metaxas [?], but there is a deeper history in the engineering literature. Foster and Metaxas’s paper was

based on the earlier work of Harlow and Welsh [HW`65] which is the origin of staggered grid methods.

In fact, many of the modern fluid simulation techniques have deep histories in engineering.

3.1.1 Eulerian Grid Fluid Simulation

Grid based simulation methods have a long history in computer graphics, starting with the seminal work

of Foster and Metaxas mentioned above. That method, and many of its successors, had difficulties with

numerical stability with large timesteps where the total energy of a system would explode. The issue

lay with the use of the use of an explicit time integrator. It was with Stable Fluids by Stam [Sta99] that

a semi-implicit integration method appeared in the graphics literature, which prevented such energy

blowups. That guarantee, however, came at the cost of significant numerical viscosity, which caused

energy to disappear from a system. Since then multiple methods have become popular in graphics to

compensate for numerical viscosity like vorticity confinement [FSJ01].

The methods mentioned above are excellent for defining fluid flow when a static domain is filled

with fluid and a few quantities are advected around by a velocity field, as is the case when simulating

smoke. However, for many graphics applications, fluid does not fill a static domain, and instead covers

a dynamic subdomain of a larger domain. For example, in a scene where water is being poured into

a glass, there is initially no water in the interior of the glass and over time the subset of the domain

16

Chapter 3. Relevant Work 17

comprised of water changes dynamically. In order to handle such a scene using a Eulerian method, some

extra work is required.

Simulating fluids with boundaries that can form waves and splash are commonly called free surface

fluids. Surface tracking technologies like the level set method [EMF02, ZOF01] have become popular

because they allow simulations to represent the interface between a fluid and air. Eulerian fluid simulation

techniques are exemplary at simulating incompressible free surface flows when properly coupled with

levelset data [BBB07]. This is due to the availability of simple differential operators for computing

the pressure required to maintain incompressibility. Purely Eulerian methods, however, are inherently

limited and have a difficult time with numerical viscosity and handling free surfaces.

3.1.2 Particle Fluid Representations

Particle-based fluid techniques are naturally amenable for simulations involving free surfaces, though

the smoothness of the generated surfaces can be difficult to maintain. There has been, however, sub-

stantial efforts in that direction like [YT13]. The most popular particle-based method used for fluid

simulation, and more recently granular simulation, is Smoothed Particle Hydrodynamics (SPH). This is

a method whose origins are in the astrophysics community in work by Gingold and Monaghan [GM77]

for tasks in compressible flow. SPH was brought into the graphics literature by the work of Desbrun and

Gascuel [DG96] for purely Lagrangian fluid simulation. Beyond representing particles being advected

through space, it represents a family of radial basis functions φti around a set of points advected through

space. Under this framework, a function f discretized at a time t would be computed by attaching

coefficients f ti “ xf, φtiy (where φti is a dual basis function for φti), therefore producing the approximation

f t “ÿ

xf, φtiyφti “

ÿ

f tiφti.

The basis functions do not inherently represent the boundary of a surface and the repulsive forces they do

generate are smooth, which naively leads to timestep restrictions and “soft” particles - in that particles

act as if they were undergoing elastic deformation. This “soft” particle property is problematic for SPH

in that it has trouble maintaining incompressibility in the long run, though work on maintaining a sem-

blance of incompressibility has been an area of active research [BT07, SP09, ICS`13]. Compressibility

is a difficult problem to solve in a purely Lagrangian context, but not too difficult to handle in Eulerian

methods, so one could imagine trying to alleviate ompression issues by combining both methods. Meth-

ods that utilize the advantages of both Eulerian and Lagrangian methods are genreally what are called

Semi-Lagrangian methods.

3.1.3 Semi-Lagrangian Fluids

Among the the most succcessful Semi-Lagrangian methods are Stable Fluids and FLIP. FLIP was in-

troduced into the graphics community by Zhu et al. [ZB05]. This method utilizes particles to advect

velocities and a grid to solve for pressure and add external forces. Although it suffers from some long

term volume loss and numerical viscosity from the large amount of interpolation that is performed , the

loss is much less than plain SPH, and the numerical viscosity is also much less than that of a standard

purely Eulerian technique. The work of Batty et al. [BBB07] extended the use of FLIP particles to

generate a levelset to generate a more accurate Laplacian operator. More recently, the Hybrid SPH


Method [RWT11] by Raveendran et al. have utilized that sort of pressure solve to guide the advection

of SPH particles to decrease compression.

Pure levelset surface tracking has evolved to utilize particles in methods like the aptly named Particle

Levelset method [EFFM02] of Enright et al. In the method they tie the signed distance function to a

family of particles situated near the interface. The particles are used to correct the levelset from the

inherent information loss that interpolation causes during levelset advection.

Traditionally Semi-Lagrangian simulation has depended on particles because managing topological

changes in fluid simulation is an arduously difficult problem to solve. More recently, meshing technology

by those like Brochu et al. [BB09] has advanced far enough that using Lagrangian interface tracking

with triangle meshing has become feasible [WMFB11]. The main disadvantage of these methods is the

necessity to invest time to develop and tune a mesh cleaning tool. Many methods have requirements

that the tessellations they use must be manifold or delauney. Another common reason why a decent

mesh cleaning is required is to make sure that the tessellated elements do not become too small or too

large. Elements that are too small restrict the size of each timestep in the simulation, which reduces

performance, while large elements can reduce the quality of the simulation.

3.2 Granular Simulation Techniques

Granular simulation has primarily been performed through purely particle-based methods, but recently

there have been more attempts to simulate them as a continuum in order to handle scalability issues

that arise from simulating too many particles.

3.2.1 Particle Granular Materials

The most obvious way to simulate dry granular materials is as a collection of rigid bodies. Although

many granular materials have a variety of anisotrophy that varies from being simple ellipsoidal shapes

like in rice grains or fairly complicateted geometries of “arbitrary” shapes such as is found in sand, a

common assumption is that in aggregate they act like spheres. As such, the most common form of

simulation of granular materials is as a familiy of rigid spheres, despite the actual geometry of what is

being simulated [YHK08]. The work of Bell et al. [BYM05] allowed for the simulation of non-spherical

particles, but they only allowed for objects represented as the union of spheres held together statically.

From that perspective simulating dry granular materials the problem becomes simply a particular use

case of rigid body simulation [KP06, Llo05], especially because the contact properties of spheres are so

easy to implement. Recently there has been some interesting work by Smith et al [SKV`12], which is

able to preserve spatial symmetries, kinetic energy, and momentum in large rigid body systems, which

extends nicely to granular media. Through their work they were able to reproduce complex emergent

patterns generated by vibrating granules in rectangular bins.

Although dry granular materials are well represented as spherical rigid bodies, cohesion is a difficult

force to represent. There has been work such as that of Lenaerts and Dutre [LD09], which utilize SPH

to simulate wet granular materials. They utilize SPH for both the fluid and the granular components

of the simulation. They utilize their preceding work on porous flow using SPH [LAD08] to simulate the

porous flow of fluid entering bodies of sand. Further work in simulating wet granular materials through

SPH was explored by Ihmsen et al. [IWT13], though it does not discuss how to do porous advection.

The work of Rungjiratananon et al. [RSKN08] uses SPH for the fluid simulation, but uses the Discrete


Element Method for simulating the forces between granules as a sort of mass-spring system generated

by the adjacency of particles in each timestep.

The Material-point method[BBS00] is a method that evolved from FLIP.TODO: find earlier refer-

ences.

Despite significant improvements on the field, solving for rigid body dynamics for large numbers of

particles is an inherently intractable task. There have been several interesting approaches, however, such

as the work of Zhu et al. [ZY10] which matches particles with a height field which allows for particles

to be labeled as rolling, interface, or static. Through that separation they only need to worry about

a narrow band of particles undergoing dynamics. The method is optimal for simulating sand being

dropped or undergoing low magnitude forces, but because there is no simulation in the lower depths of

material, the method is not amenable to other situations.

3.2.2 Continuum Granular Materials

The first explicit use of continuum methods for simulating granular materials was in the original graphics

paper for FLIP by Zhu et al. [ZB05]. The method implements friction through a linear scaling of

tangential velocities and clamping them to a non-negative value. They even support cohesion in their

model, but mainly to remove some seepage issues that were a side effect of using such a simple friction

model.

To solve for an even “stronger” form of incompressibility than what is used in FLIP, Narain et

al. utilize the Unilateral Incompressibility Constraint [NGCL09] in an effort to guarantee that the

constituent particles will not intersect with each other. Their application for that result was not for fluid

simulation, but rather for simulating large numbers of independent agents such as crowds navigating

through terrain as sophisticated as the interior of a building. They were able to handle large quantities

of agents because, although the agents individually only respond to local stimuli, global flow issues are

handled through a linear complementary problem, which enforces a density constraint for each cell in

a regular grid. The constraint maintains an L2 optimal set of forces such that no cell can have higher

density of agents than that prescribed by the optimal sphere packing density.

By setting these particle-agents to advect in the same fashion as a FLIP fluid simulation, and by re-

placing the viscosity computation with a constrained energy minimization simulation of friction, Narain

et al. were able to convincingly simulate granular materials as a continuum flow [NGL10]. This con-

strained energy minimization problem is solved with a quadratic programming problem, where the yield

criterion between shearing strain and pressure formulated approximately through a set of constraint

hyperplanes.

3.3 Porous Flow

Porous flow, also called Darcy flow, lacks the visual intrigue that vortices of smokes and splashing of

water, but has serious consequences on the rheological behavior of materials, as in sand and paper.

Although we have mentioned some methods that can support cohesive granular materials, and even wet

ones, they do not contain a proper discussion of porous flow. Some of the work like Zhu et al. [ZB05]

and Ihmnsen et al. [IWT13] simulate cohesive granular materials, but are silent on the fluid causing the

fluid to be cohesive, while in the work by Rungjiratananon et al. [RSKN08] water particles are removed

upon contact with granular particles to accumulate “wetness” on each granular particle. In that method,


the “wetness” of individual granular particles are propagated only when a particle becomes “over-wet”,

simulating that a granular particle only propagates wetness once it becomes saturated with water. They

also assume that capillary forces are insignificant compared to gravity.

Within Graphics, the study of Darcy flow has not received much attention, only really being men-

tioned in the work of Lenaerts et al [LAD08] that derived everything through SPH. We found that the

book by Bear [Bea72] is a very good resource for the general flow of the phenomenon, including some

insights on the nature of capillary pressure and individual models for different pore topologies in porous

media. Furthermore, the work by Hirani et al. [?] discuss how to solve the problem of Darcy flow using

Discrete Exterior Calculus.

Chapter 4

Overview of Simulation Methods

In this chapter we provide a high level description of the core components of our method. The method

combines two independent simulations: one for the fluid component and one for the granular material.

We then connect the two simulations by mapping pertinent physical characteristics between the two

simulations as parameters. Through the synchronization of these parameters in the respective simulations

we obtain a coupling between the two simulations that is visually convincing.

In order to utilize these parameters in our simulations of granular materials and newtownian fluids

we had to augment existing techniques for both the fluid and granular simulations, which we will soon

discuss. This chapter will walk through the process of augmenting and combining simulation methods

through a series of steps. We will begin by describing a novel method for computing a Hodge-Star from

a levelset, which is a crucial component in everything that follows. From there, we will discuss a method

for simulating a free surface fluid immersed in a porous body where porosity is variable over both time

and space. Independently we will describe a method for simulating a cohesive granular material where

the cohesion can change over time and space. Finally we will discuss how to combine the two techniques

to provide a method for simulating the interactions between a fluid and granular body.

Before we enter the core of our work we will provide a light discussion on our choice of discretization.

4.1 Operator Choice

We utilize operators from exterior calculus, as described in A. The primary reason for utilizing this

extra structure is that it makes implementation much simpler, while at the same time making the theory

more general to other types of discretization. This ease of implementation is due to the small number

of operators that need to be implemented and the simplicity of those individual operators, which makes

it possible to create a full working system with less code to implement and debug. Furthermore, those

operators are amenable to a wide variety of other discretization techniques beyond the one that we chose.

This is because they only depend on having a cell complex with some reasonable concept of a dual mesh

attached. This mathematical language is also dimension agnostic, which implies that the algorithms

defined here have easily extensions to even higher dimensions with minimal effort, which accelerated our

conversion from 2D to 3D code.

21

Chapter 4. Overview of Simulation Methods 22

4.1.1 Cubical Complexes

Although the following theory is trivially generalized to other sorts of complexes, we will be only using

cubical complexes. We choose to only discuss cubical complexes because although other structures, like

simplicial complexes, are from a theoretical perspective simpler, we want to show that our method is

not only equivalent to other existing grid-based methods and can even be easier to implement than

them as well. We include some useful implementation details in Appendix B which defines the cubical

structure somewhat rigorously and presents the simplifications that cubical complexes have over generic

cell complexes.

4.2 Levelset Based Hodge Star

As discussed in the Appendix B the Hodge Star operates as a way to map between k-forms and pn´ kq-

forms and allows for the definition of an inner product on differential forms. For simplicity of imple-

mentation DEC uses the diagonal Hodge Star operator, a sort of mass lumping that guarantees that

between a k-cochain and a pn´kq-cochain that the evaluation of a k-form and its respective pn´kq-form

on the k-cochain and pn ´ kq-cochain produce the same value. In order to guarantee an isomorphism

between k-cochains and pn´ kq-cochains in the discrete setting, DEC introduces a dual mesh, for which

the Hodge Star defines a correspondence between the primal mesh and its dual mesh. In general the

functionalities available to the primal mesh are identical to those available to those of the dual mesh

and the difference comes down to the topological and geometric features of the choice of primal and dual

meshes.

For instance, the vertices of the dual mesh are usually defined by the circumcenters of simplices,

which is created by attaching a vertex at the circumcenter of every n-cochain, a dual edge between dual

vertices for which their corresponding primal n-cochain share a pn ´ 1q-cochain, and so on. The above

construction leads to a trivial computation for the discrete boundary operator of the dual co-chains:

Bk`1k “ pB

n´kn´k´1q

t

where Bk`1k denotes the dual boundary operator and ¨t represents the transpose operator. In later sec-

tions we will not distinguish between primal and dual operators because the choice of storage between

the primal and dual mesh is generally arbitrary. The Hodge Star operator, however, is the main corre-

spondence between the primal and dual meshes so we will still talk about primal and dual meshes for

the remainder of this section.

The discrete Hodge Star maps a k-form ωk sitting on k-cochain σk to a pn´kq-form ‹ωk on pn´kq-

cochain σn´k. We compute the value of the Hodge star by first recalling its definition:

1

σk

ż

σk

ωk “1

σn´k

ż

σn´k

‹kωk.

By using the integrated quantities as the discrete value for the forms and by lettingş

σn´k ‹k¨ “ ‹k ş

σn´k ¨


we obtain the diagonal Hodge Star:

1

σk

ż

σk

ωk “1

σn´k

ż

σn´k

‹kωk

1

σkωk “

1

σn´k‹kωk

σn´k

σkωk “ ‹kωk.

Standard DEC is designed around having a static primal and dual meshes but we extend the concept of

the diagonal Hodge star to describe more dynamic situations.

Our method operates by maintaining a static dual mesh (in this case a regular cubical complex) and

extrapolating a dynamic primal mesh extrapolated from a levelset. A standard problem with levelsets

is that they do not have any inherent information on the input geometry within a single cube, so the

particular geometry inside of a cube is underdetermined. We therefore do not generally have enough

knowledge to explicitly generate the primal mesh, and in fact take advantage of the fact that we have

some unknown information to virtually pick the interior data that satisfies our wants. In particular we

use the levelset to determine the Hodge star directly from the levelset using linear interpolation schemes.

As levelsets are scalar fields, it is natural to store the levelset on the primal or dual 0-forms, and for

each of these types of levelsets there are different ways to extrapolate quantities.

4.2.0.1 Length of edges

For a levelset φp stored on the primal grid we can compute the length of the primal edge e “ vi, vj from

the levelset directly. If φp ď 0 at both points then volpeq “ ∆x and if φp ą 0 on both then volpeq “ 0.

WLOG, assume φppviq ď 0 and φppvjq ą 0 and we can set:

volpeq “ ∆xφppviq

φppviq ´ φppvjq

which is, under the assumption that φp|e is linear, ∆x scaled by the fraction of the edge such that φp ă 0.

If the levelset φd is stored on the dual grid, the length of a primal edge requires a bit more effort. If

we assume that levelset is bilinear within each dual cube we can average the values of φs at each of the

dual vertices to generate a primal φ1s, for which we can use the above procedure for generating primal

volumes on primal grids.

The obvious analogues work for computing edge lengths on a dual mesh.

4.2.0.2 General Volumes

In general, for higher dimensioned edges there is more freedom in how one can compute volume from

a levelset. As in the case of edge lengths, for dual elements we have to take averages to translate

the primal levelset to a dual levelset. We assume that objects are piecewise linear and utilize the

volumes of the triangles/tetrahedrons generated by evaluating a square/cube respectively using marching

squares/cubes.

Once we have the volumes, we can generate the Hodge star operator between the primal and dual

elements through dividing the primal and dual volumes and vice versa.


4.2.0.3 Dealing with Degeneracies

One complication with these volumes is that we may have degenerate elements (elements with 0 vol-

ume). When these entries are used to compute the Hodge star, the resulting operator is no longer an

isomorphism between the primal and dual forms, which is a fundamental property of the Hodge Star.

This, however, is not an issue, as the places where the Hodge star is degenerate are precisely the places

where computation is unimportant. In our problems our only concern is to solve for pressure inside the

fluid domain, which is definitely outside of the solid domain. The Hodge star on that restricted region

is guaranteed to be nondegenerate because the levelset of a solid is always disjoint from the levelset of

a fluid, and the levelset for a fluid is always inside the levelset of a fluid. By zeroing the Hodge star

when we get zero or infinite volume ratios we obtain a projection operator on forms that projects into

the restricted space.

This projection, along with some numerical considerations, will create some additional restrictions on

how we set up the system. One important consideration is that we will not be projecting the boundary

operator to the projected space, which will result in an underdetermined system unless we take special

care in whether to store our quantities in the primal and dual meshes.

4.2.1 Pressure Projection

Of course the final goal of all of this theory is to produce a satisfactory system of equations to perform

the standard pressure projection step in fluid simulation. By storing pressure p as a primal n-form and

velocity u as a dual 1-form we we can obtain the amount of “pressure”, we can decompose u by using

the Hodge decomposition to obtain

u` d‹ p “ u

where u is the summation of the two remaining terms from the Hodge decomposition: u “ δβ`γ. Since

δδ “ 0 and the fact that δγ “ 0 for harmonic functions, we see that δu “ 0.

Because u is in the kernel of δ we can do the following:

u` d‹ p “ u

δu` δd‹ p “ δu

‹d‹ d‹ p “ ‹d‹ u

d‹ d‹ p “ d‹ u.

By storing the dual of pressure as p we get the weak Laplace-deRham operatorciteabraham1988manifolds:

d‹ d‹ p “ d‹ u.

d‹ dp “ d‹ u.

Lp “ d‹ u.

This produces a positive symmetric semidefinite linear system, which can solved efficiently using pre-

conditioned conjugate gradient, with a modified incomplete Cholesky factorization as the preconditioner,


following [BBB07].

From there one can easily find a value u by

u “ u´ dp

which is the L2 projection of u into the space of gradient free fields.

4.2.2 Free Surface Boundaries

We require one further addition to compensate for the force of pressure on the fluid boundary, which is

taken care of by using the ghost fluid method mentioned in [EMF02]. With a free surface boundary the

volume of fluid is not contained and the pressure from the air must be taken into account. The standard

solution is to set a Neumann boundary constraint that the pressure at the boundary is 0, which is what

we do. However, applying the boundary condition with levelsets is difficult because the interface is not

precisely defined on a grid boundary, and so a method such as the ghost fluid method is necessary. The

ghost fluid method uses Neumann constraints to extrapolate values of a scalar quantity right outside

the boundary. This is particularly relevant in our case, where the quantity is pressure and the boundary

value is always 0.

If we use a linear extrapolation of pressure from a cell-center pi that is inside to generate the value

of a cell-center outside pj , such that φpxiq “ ´θ we see that the values must satisfy

0 “ p1` θqpi ´ θpj

which provides us with

pj “1` θ

θpi.

If we look at the effect of the exterior derivative operator d on pi, pj , we see that on the face they share

we get, up to sign dependent on orientation,

dp|Ω “ pj ´ pi|Ω “1

θpi “

1

volpeijq“ ‹l|Ωp

where ‹l|Ω is the Hodge Star restricted to the boundary. If we take d‹s of the operator to generate

a full Laplacian, notice that by definition of the situation only the cell storing pi will only have one

side of the pertinent axis nonzero, so the d‹s simply maps the the face value to the vertex cell-center.

By noticing that we are purely adding terms to the system, the result of the linear extrapolation is the

following system:

pLl `‹s ‹l |Ωqp “ d‹l u,

It is easy to convince oneself that this is still a positive semidefinite symmetric system which is advan-

tageous for numerical solvers.


4.3 Darcy Flow

To extend the above method to support flow in a porous medium, also called Darcy flow, there are two

key insights. The first is that the porosity can be treated as a scaling of the permeability through a

plane, which leads to a porosity-based Hodge Star. The second is traversing through a porous medium

mutes the momentum of a porous material, and so the velocity from a previous step has no bearing on

the velocity of the current timestep. In fact, the velocity of fluid in a porous medium is determined by

velocity of the porous medium itself, the pressure effects to maintain incompressibility, and the force

capillary pressure.

4.3.1 Porosity

In our simulation technique we use a scalar porosity function ρ across the whole domain, with the value

1 specifying that fluid can fully fill a region and 0 signifying that the body in that space is completely

filled. By defining porosity as a scalar function across over our entire domain we obtain a representation

that allows for variable porosity throughout both time and space. When ρ “ 1, there are no obstructions

to fluids flowing through that neighborhood, so any fluid in the region tx : ρpxq “ 1u can be represented

as a standard free surface fluid. However, when porosity is below some threshold we assume that the

fluid in the region will be under a porous flow regime and will have the same velocity as the bulk material

surrounding it.

4.3.2 Porosity Based Hodge Star

We model the porosity as the percentage of a volume that can be filled with fluid material. The amount

of available volume is the sum of available volume between open air and the volume available in porous

solid materials. We assume that all solid material in a region is porous with the same porosity ρ found

in a grid. That is, the solid volume in a region of space V is

p1´ ρqvolφspV q.

The total amount of fluid that can fill a volume is then defined by a scaling of the solid volume and the

remaining available space:

ρvolφspV q ` pvolpV q ´ volφspV qq.

If we modify the Hodge Star operator to take into account this modified fluid volume we obtain a Hodge

Star that represents the porous medium. The previous assumption could be removed by keeping track

of the porosity of each constitutent object in a solid levelset and subtracting the solid volume filled by

the materials appropriately. By using the above formulae we can define modified versions of the fluid

Hodge Star operator, and apply the same pressure solving technique as before.

4.3.3 Capillary Forces

Our capillary forces are derived by the difference in potential between the wetted and dry phases of our

granular material, in our case water and air.

There are various models that assume different microstructure geometries [?] but we have chosen

to go with a simplen?¨n´1 relationship to map volume to surface area due to its simplicity. Most


graphics fluid methods assume that air is really a vacuum, and we will continue with that except for

when considering capillary potential, which depends on the existence of air. In particular we do not

consider wind blowing on the sand, leaving this topic to future work.

We chose to represent this capillary potential Φ by setting the potential on air-surface cells to be

0 and on fluid-surface cells to be a function of the fluid density per cell, while taking into account the

surface area of porous material available. By using the previously mentioned approximation between

volume and surface area, and by taking into account that the fluid-solid interface cannot be more than

the minimum surface area of the two, we obtain

Φ “ n

b

minpV olf pV q, V olspV qqpn´1q

.

Though this is an extremely crude approximation of the potential between the two areas, it is not the

focus of this work. A correct model for the solid porous region would require analysis on a per-object

basis and would likely exceed our needs for our phenomenological purposes. A more careful scheme is

left for future work.

4.3.3.1 Maintaining Particle Density

A basic FLIP implementation has no mechanism for managing the density of fluid. This is very important

when the porosity of a region changes such as when a sponge is squeezed. Although FLIP generally

has satisfactory volume preservation properties, the particles in a FLIP simulation have a tendency to

clump together. Once they do, they rarely ever separate, which causes volume loss. [NGCL09] utilized

a lightweight algorithm to separate particles in each timestep in their UIC simulation, which we likewise

apply in our FLIP simulation. The scheme applies symmetric pair-wise position modifications in an

attempt to cheaply remove as much self-intersection as possible. Im a general flow of particles such a

scheme fail in when multiple particles converged, so these corrections would not work. However, because

fluid particles move in a relatively divergence-free fashion, the piecewise corrections become the sufficient

nudge required to prevent the vast majority of intersections. In fact, we applied this pairwise scheme on

a normal FLIP simulation and observed almost no particle self-intersection and significantly less volume

loss without any seemingly different phenomenology.

This method of particle separation does not require both sets of particles to be of the same size,

though one could imagine making the repulsion match the interacting particles’ radii. Because the fluid

simulation method described so far does not explicitly take into account the density change that occurs

when a parcel of fluid goes into a porous material, the density of fluid inside a porous region could be

equal to the density outside, which is not physically plausible. Because the porous material fills space,

the fluid must diffuse and fill more volume at a continuum level. In order to compensate for the required

density loss we change the radius of each fluid particle in order to give the particle a constant volume

while immersed in a porous medium. Specifically, given a particle with air-based radius r in a porous

medium of porosity ρ, we satisfy the equality

4

3πr3 “ ρ

4

3πr3

with the new radius being r “ 13?ρr. In higher dimensions, of course, the radius would be scaled by 1

n?ρ .

We utilized spatial hashing in order to accelerate the performance of the pointwise comparisons. By


creating cell-center based storage grid that could hold vectors of grids we stored particles on these grids

and performed the pointwise comparisons between every particle in a single cell with all of the adjacent

cells.

4.4 Cohesive Granular Materials

The rheological behavior of wet granular materials is a generalization of the behavior of granular materials

that stick together. We therefore first discuss how to simulate a granular material that has cohesive

forces sticking the particles together. Although cohesion holds particles together at a per-particle-pair

isotrophic force, at a continuum level the effect of cohesion on a bulk of granular material results in both

isotropic and deviatoric strains.

4.4.1 Isotropic Strain

The theory behind how to deal with isotropic strain is quite simple: each particle induces an attractive

force to every other particle to which it is connected. In practice, we utilize the most rudimentary

method possible: we introduce attractive forces between two particles in order to introduce particle

level cohesion. We generate this particle-wise force during the collision detection step in order to take

advantage of the existing distance comparisons. We add cohesive forces to particles a of distance less

than 1.1 ˚ radius through a radial cube spline function once collisions have been dealt with.

The simple, explicit scheme described above is too naive to support more delicate overhanging be-

haviors, but it appears to be sufficient for small scale clumping and for preventing the boundary of wet

material from eroding. The use of a more sophisticated method would be prohibitive at the scales in

which we are interested, but would definitely be interesting further work.

4.4.2 Deviatoric Strain

The deviatoric forces are a direct result of the complex structure generated by a network of cohesive

particles pulling on each other. This results in a force to resist shearing, which increases the amount

of shearing strain required to make a bulk yield. This is why the yield criterion utilizes cohesion as an

added scalar factor to the right hand side of the yield criterion:

s ď?

3αp` c.

4.4.2.1 Discretization

Since the deviatoric effects of cohesion are primarily an artifact of homogenization of a bulk of granular

material, we can only discretize it at a continuous level. We therefore only deal with these effects at the

continuum level of our representation.

The approach we take at the continuum level is through an extension of the original dry granular

material method of Narain et al., but with some further modifications to support the effects of cohesion.


With cohesion, the system of discrete formulation we obtain is

sTA2s` bT2 s ě 0 (4.1)

psmax ` cq ě s ě ´psmax ` cq (4.2)

(4.3)

where A2 and b2 are defined as they were before at 2.8 and 2.9 respectively.

As used by Narain et al., we solve this system of equations by using a customized implementation

of Modified Proportions with Reduced Gradient Projections, which we will discuss in more detail in the

technical details section in order to handle the above linear complimentary problem.

One artifact that appears is that any cell that has nonzero cohesion is connected in this linear system.

Cohesive cells with few particles act as if they are filled with some cohesive aether and prevent actual

particles from moving. Because of this issue some care must be taken to reduce the amount of cohesion

in order to prevent cohesive forces from being generated by cells with few particles. We choose to not

include cohesion in cells where the particle density is below some threshold that we manually tuned.

4.4.3 Using Levelset Hodge Star for UIC

Similar to the original paper on the Unilateral Incompessibility Condition [NGCL09] we switch from using

the Laplacian dT d to using the weak Laplace-deRham operator dT ‹ d to take into account the amount

of solid available in a region. In the original UIC discussion the population density is considered, but

we utilize our existing code for computing the Levelset Hodge Star to approximate the granular density

in a cell.

4.5 Wet Granular Materials

The final step is to merge our above methods for free surface fluid simulation with porous materials

and cohesive granular materials into a single simulation method for a coupled simulation of fluid and

granular materials.

As we have mentioned before, we create this coupling by changing the porosity of space in the fluid

simulation and by changing the cohesion of particles in the granular simulation. We do this through a

fairly simple connection at the continuum level and through a bit of care at the particle level.

Porosity is particularly easy to define; it is simply the fraction of volume, per unit volume, that is

free:

ρ “volpV q ´ volsandpV q

volpV q.

On the other hand, the cohesion forces introduced by the fluid into the sand simulation is something

we need to model. In reality computing the cohesion requires analysis on the aggregate behavior of

particles being held together with water without any consideration of contact geometry or material

properties. For simplicity, we utilize a simple linear function of the fluid volume fraction in a cell as well:

φ “volpV q ´ volfluidpV q

volpV q.


4.5.1 Ordering Operations

There are two crucial components of each simulation method which require careful ordering of operations.

The first is that we need to pass the coupled quantities between the two simulations, and the second

is that particle advection needs to be performed with care so the fluid in a mass of granular material

moves with the granular material.

We chose to follow a simple approach for the passing of coupled quantities: we generate the porosity

and cohesion grids at the very beginning of each timestep. This choice seems reasonable to us because

the various coupled quantities only depend on particle densities, and those densities only change during

advection, which is at the end of each timestep. Thus throughout the evaluation of a timestep, until

the advection at the end, the coupled quantities are consistent with the simulations from which they are

derived.

4.5.2 Advecting quantities

For advection we look at the momentum stored in each of the velocity grids to get a velocity field for

the whole system. In general, we maintain separate velocity grids for each simulation, but for only the

advection part of FLIP/PIC we pull velocities from this merged grid:

u “mfluidufluid `msandusand

mfluid `msand.

For regions where there is only granular material or only fluid, the velocity maintains the same value as

the respective velocity field, and in merged regions we obtain a unified velocity field.

4.6 Summary

We began this chapter by defining the Levelset Hodge Star, which allowed us to reinterpret the pressure

solve from of Batty et al.[BBB07] and the first order ghost fluid method [EFFM02] in the language of

Discrete Exterior Calculus. This reformulation allowed us to then generalize the Levelset Hodge Star to

support porosity, which led to a porosity-aware pressure solve. We then developed a model for capillary

pressure and a method for obtaining reasonable particle densities in porous materials. With those three

components we were able to develop a method for simulating fluid flow in porous materials.

We then developed local and global modifications to the granular simulation method of Narain et

al. [NGL10] to support cohesion. Finally, we combined the two simulation methods to create a technique

for simulating the interactions between a fluid and a granular material.

Chapter 5

Technical Details

5.1 Introduction

In our opinion the most challenging components of the techniques we discuss in this thesis are the

numerical solvers required for solving systems efficiently. The systems we solve all utilize banded,

symmetric, positive semidefinite problems. We know that to be true because all of our matrices are

generated by ATA or ATDA where D is a diagonal matrix, where A is a sparse banded matrix. Therefore

we have a perfect setup for using Krylov subspace methods - namely the conjugate gradient method.

A great reference for both solving linear systems and for doing conjugate gradient is Robert Bridson’s

SIGGRAPH 2007 notes [BMF07] so we will not go over the details here. In particular, it discusses how

to use MIC0 preconditioned conjugate gradient which is one of the best choices for solving linear systems

composed of the types of matrices we encounter.

For our systems that are quadratic programming or linear complementary problems, however, stan-

dard conjugate gradient doesn’t directly work because of the boundary constraints. Therefore we use

a form of conjugate gradient that can handle linear inequality constraints, namely MPRGP (Modi-

fied Proportioning with Reduced Gradient Projections). If the reader needs a refresher on quadratic

programming or linear complementary problems please refer to Appendix C.

It’s worth noticing that in the cases where our systems aren’t positive definite we can use the minimal-

norm solutions to the systems of equations for two reasons. First, because we only use the derivatives

of our solutions, adding a constant to the coefficients doesn’t change the results.

Second, because our levelset Hodge star is applied on the solutions, extraneous information from

entries represnting data outside of the levelset are zerod out and ignored in the min-norm solution.

Because MPRGP is a fairly recent algorithm [DS05] and we need to modify it, we will discuss it in detail

for completeness.

5.2 Modified MPRGP Quadratic Programming Solver

Like [NGCL09] we utilize a version of MPRGP with modifications for handling our constraint manifolds.

For satisfying the Unilateral Incompressibility Constraint we can utilize the algorithm without modifi-

cations, because the base solver requires a constraint manifold of the form tx “ pxiq|x P Rn, xi ě ìu.

However, for solving friction, the constraint manifold is tx “ pxiq|x P Rn, |xi| ď ìu. We will now provide

31

Chapter 5. Technical Details 32

some background knowledge for solving these sorts of systems.

5.2.1 Convergence Criterion

Having a good stop criterion for LCP and mLCP problems can be tricky because of the complimentary

condition. For LCP residuals are also not useful because the constraints will usually force the solution

to be far away from where the residual goes to 0. A particularly nice and easy error metric for a normal

LCP system is given by

E “ÿ

i

p|ziwi| ´minpzi, 0q ´minpwi, 0qq

where our current solution z and residual w are punished for negative values and also for being far from

complimentary.

For mLCP we have a more sophisticated stop condition because we need to confirm that the residuals

ρka, ρkb , ρ

kc go to 0 where they are defined by

ρka “ Auk ` Cvk ` a

ρkb “ minitvi, pC

Tuk `Bvk ´ wk ` bqiu

ρkc “ maxit0,´pCTuk `Bvk ´ wk ` bqiu

This leads to an error function

E “ max

ˆ

ρka1` a

ρkb1` b

ρkc1` b2

˙

We will use this criterion as our stopping criterion in the algorithm defined below.

5.2.2 Projection Methods

One common technique for handling the nonlinearity of LCP problems, which is how MPRGP operates,

is to explore specific submanifolds of the entire constraint manifold. The usual restriction is that, given

that the constraint manifold is a set of axis-aligned planes, to only modify a subset of the variables at a

time. The set of variables that are currently active is called the active set.

5.2.3 Notation

Before we begin, with the details of the algorithm, we will first introduce some notation that will be

used significantly in the algorithm. The manifold solutions that satisfies our constraints is defined as Ω.

Let g be the current descent direction and let β and φ be vector functions defined as follows:

φipxq “ gipxq@i P Fpxq φipxq “ 0@i P Apxqβipxq “ 0@i P Fpxq βipxq “ mintgi, 0u@i P Apxq.

where

• φ,the free gradient operator, describes the proportion of g that is not in a constrained dimension.


• β, the chopped gradient operator, describes the proportion of g that goes in a constrained dimen-

sion, though modified as to not break the constraint.

Together they form νpxq “ φpxq ` βpxq, which is 0 precisely when the KKT conditions are met.

5.2.4 Steps

The algorithm works by picking one of three step choices each iteration: a conjugate gradient step, an

expansion step, or a proportioning step. The conjugate gradient step is precisely the standard conjugate

gradient step, while the other two steps are corrections to take into account the constraints. The choice

of step type in a given iteration is made by determining the relative magnitude of the current descent

direction that breaks through Ω through.

The simpler of the two correctional steps is the expansion step. It’s the reaction to when conjugate

gradient is about to break a constraint: it moves the current solution in the same direction conjugate

gradient would normlly go, but stops right at the constraint boundary. It’s derived quite simply:

If we let PΩ be a projection operator back onto Ω and α some minimal step size parameter set in the

interval p0, M´1s we set the expansion step to be

xk`1 “ PΩpxk ´ αφpxkqq.

This projection operator can be made more explicit by defining a reduced free gradient operator φ which

will set xk`1 to activate the constraint in the step. It’s defined by:

φpxq “ mintpxi ´ liqα, φiu

which, because xk P Ω, allows us to write the projection operator in terms of

xk`1 “ PΩpxk ´ αgpxqq “ xk ´ αpφpxkq ` βpxkqq.

The logic for whether to use the proportioning step is checked before the conjugate gradient step

length and is used to push away from useless active constraints as much as possible . This is done with

the inequality

βpxq2 ď Γ2φpxqTφpxq

with a parameter Γ in p0, 1s to determine the proportion of φ the gradient is allowed to face into Ω. If

the proportioning step is chosen, it does one round of conjugate gradient using β as the gradient for that

step.

5.2.5 Performance

Conjugate gradient, as a Krylov subspace method, depends on iterating through a sequence of matrix

multiplies off of an initial vector v tAkvu. This sequence of matrix products conflicts with projection

operators because modifying only the active set destroys information of the subspace explored by the

multiplication, thus ruining the convergence properties of the method.

The way that MPRGP deals with that fundamental issue with conjugate gradient is that it only uses

a projection when the algorithm things that there will be sufficient gains by doing so. In our experience


Algorithm 1 Modified Proportioning witih Reduced Gradient Projections

Set r Ð Ax´ b, pÐ φpxqwhile νpxq ą ε do

if βpxq2 ď Γ2φpxqTφpxq thenαcg Ð rT ppTApαf Ð maxtα : x´ αp P Ωuif αcg ď αf then

Conjugate Gradient StepxÐ x´ αcgpr Ð φpxq ´ αcgAppÐ φpxq

elseExpansion StepxÐ x´ αfpr Ð r ´ αfApxÐ PΩpx´ αφpxqqr Ð Ax´ bpÐ φpxq

end ifelse

Proportioning StepdÐ βpxqαcg Ð rT ddTAdxÐ x´ αcgdr Ð r ´ αcgAdpÐ φpxq

end ifend while


this doesn’t happen very frequently, mostly as the algorithm nears convergence on a solution. That sort

of experimental evidence led us to try applying MIC(0) preconditioning on the conjugate gradient steps,

which was effective.

Chapter 6

Results

In this section we will briefly describe our implementation and then continue to discuss some visual

results of our algorithm.

6.1 Implementation Details

We implemented our method using C++ using Eigen as our matrix and vector storage mechanisms. We

utilized C++ templates rather heavily in order to define the various grid types by the dimension of the

forms that they held as well as to define and store the various differential operators. In general we found

that template metaprogramming significantly improved the debugging experience because compilation

errors would report bugs expediently rather than at runtime. This was especially convenient for storing

quantities like the offsets used between different grids for their origins and dimensions.

We implemented our own linear and quadratic solver libraries, though our MIC0 cholesky factorizer

was a translated version of Robert Bridson’s code around Eigen’s sparse matrix implementation. For

generating scenes we also utilized Bridson’s fast poisson-disk sampling scheme [Bri07].

6.2 Porous Results

Our method for simulating porous fluid can be seen in the sequence of images Figure 6.1, which displays

several different phenomena related to porous flow. On the top left there is a circular fluid source, which

we will call the emitter, that emits fluid to the right. To the right of the fluid source there is a porous

wall that acts as a barrier that limits the flow of fluid from the left void to the void. Fluid still, however,

does seep from the porous wall as the wall is saturated with fluid particles (Figure 6.1c).

The seeping action is particularly visible at the point where the emitter shoots water into the wall.

Some of the seepage is caused by the fluid pushing itself into the porous body, as can be seen in

Figure 6.1b: the velocity field inside the porous region indicates that the fluid is trying to push through

the porous region. This phenomenon is caused by the porous pressure that we compute. The computed

pressure makes penetration into the porous medium more difficult for the fluid than flowing back into

the voids, so although some fluid is being pushed into the porous areas, splashing occurs and fluid flows

parallel to the porous body. This splashing behavior is already visible in Figure 6.1b as the fluid begins

to rush to the left after hitting the wall in a clockwise fashion even though the area is not yet saturated.

36

Chapter 6. Results 37

(a) (b) (c)

Figure 6.1: Fluid being spilled into the junction between a porous block and a porous wall. The rednessof the background grid implies the porosity, the white boundary defines the fluid boundary, and theturquoise lines label the velocity field

Furthermore, in Figure 6.1c both voids have vortices that avoid the porous region because of how the

pressure solve restricts the fluid from entering that region.

The effectiveness of using pairwise particle separation in standard FLIP fluid simulations can be seen

in Figure 6.2. The fluid particles have very little self-interesction except at the boundary, which is a

difficult region to prevent intersection because the particles have less movement options. Intersecting

particles on straight boundaries can only move along those boundaries, which means that there is only

a one-dimensional axis for which these particles can move. These sorts of density-based issues are much

more noticeable in the 2D case than in the 3D case, where in general a larger proportion of particles will

have more than one axis of available movement. Our method of modifying the particle radii according to

the region’s porosity is also visible in Figure 6.2: the particles in the porous areas have an appropriately

lower density than in the voids.

6.2.1 2D Rendering Artifacts

Figure 6.2 illuminates two types of rendering artifacts that cause issues in our 2D boundary rendering

that are easily fixed. These artifacts are the duals of one another: sometimes particles disappear and

sometimes voids appear in fluid bodies. In order to avoid clutter when rendering sand particles we

utilize a boundary based rendering approach. It is created by running the marching squares algorithm

on the levelset we use for computing the Levelset Hodge Star. Because the grid is purposely coarser than

the particles, sometimes the particles travel in paths that make them invisible to the marching sqaures

algorithm. The algorithm depends on sign flips on edges to determine vertices and edges to render, so

because there are no sign flips nothing is rendered. The inverse issue is that sometimes there are not

enough particles in a sampling location, which then causes a void to be rendered although no effects like

cavitation are being simulated.

The solution to these artifacts would be to increase the resolution of the levelset generated for


Figure 6.2: Figure of fluid particles rendered in the same simulation as Figure 6.1


(a) (b) (c)

Figure 6.3: Water and sand being emitted in near proximity

rendering, which is a trivial operation to do but we did not think it would be necessary to prove the

effectiveness of our method for the purposes of this thesis.

6.3 Combined Fluid and Granular Results

Our method successfully increases the steepness of sand piles and increases sand’s resistance to movement.

We focus on the steepness of piles generated because that is a useful characteristic for determining

the effectiveness of cohesion. As the cohesion of sand particles increases, sand becomes capable of

achieving steeper angles because particles hold each other together. It’s this effect that makes the sort

of overhanging behavior required to make sandcastles possible. In fact we are capable of producing a

small amount of overhanging behavior.

Figure 6.3 shows a scene where sand dropped from the center of a scene and water is dropped to

its left. The cohesive effects of the water are clearly visible by Figure 6.3c as the sand that dropped

forms a significantly more stable pile on the left side than on the right side. Because we wanted to

generate higher piles we placed our emitters fairly high up, which caused impulses onto the sand pile.

These impulses reduced the steepness of the piles generated. Furthermore, we had difficulty creating

a constant stream of sand because particles coming out of the emitter tended to interact with other

particles, causing the particles to come out in a spray.

In order to see the effects of sand falling into an existing cohesive region, we dropped sand into

an existing pool of water (see Figure 6.4). In this simulation we were able to see that for very light,

completely submerged piles, we were able to obtain very steep hills. The initial impact passed the

momentum accumulated by the falling sand particles to shoot fluid particles into the air as can be seen

in Figure 6.4e. Once the sand had risen above the water it once again formed a shallow pile and the

submerged portion obtained a more shallow slope as well.

After we had run the simulation behind Figure 6.4 for a while tried turning back on the water emitter,

which resulted in what we show in Figure 6.5. The configuration became reminescent of the porous scene

for Figure 6.1 as the sand began partitioning the water into two parts. Some of the momentum from the

fluid seemed to pass into the sand and the pile began leaning away from the water and some overhanging

behavior emerged.


(a) (b) (c)

(d) (e)

Figure 6.4: Sand being dropped into water.

(a) (b) (c)

Figure 6.5: Water and sand being emitted in near proximity

Chapter 7

Concluding Remarks

Before we conclude this document, we shall first discuss some potential areas of future research for the

methods we have described in this document.

7.1 Future Work

The most blatant omission of this work is that it doesn’t have a working 3D counterpart, which we will

look into in the near future. Beyond that there are still several issues with the method that need to

be taken care. Many of the modeling assumptions in this paper were rudimentary and linear operators

were chosen instead of measured quantities that correspond to actual material properties. Some of these

modeling assumptions include the effect of wetness on cohesion could be done without changing the

performance of the system because they would simply be changes to different grids.

Our method is not necessarily limited to only ball-like shapes and it would be interesting to see

whether the simulation could be extended to more complicated or even heterogeneous particle shapes.

That sort of exploration, however, would require some improvement on the stability of the system.

7.1.1 Stability Issues

We experienced a significant amount of stability issues with our sand simulation method. This was

because particles bouncing into each other would significantly alter the velocity field in an area, which

caused widespread vibrations that didn’t stabilize. Using an alternative medium for connecting the

particles to the grid like PIC instead of FLIP might have fixed some of those issues, but it would have

made the dry sand lump together a bit more than desired. However, PIC would have been an excellent

choice for advecting sand that was cohesive enough. Since the bulk of this work was done the material

point method has gained some popularity in graphics [SSC`13] and would be appropriate for replacing

the current granular simulation method.

7.2 Conclusion

In this work we have presented several novel contributions to the area of physically-based simulation in

computer graphics. In order to achieve this result we have delved into the theory behind existing fluid

simulation results for by reinterpreting their operators in terms of Discrete Exterior Calculus. Using

41

Chapter 7. Concluding Remarks 42

this reinterpretation we have extended those results to support porous fluid flow without much in the

algorithms. We have also extended a dry granular material technique to support cohesion between par-

ticles at multiple scales. Finally we have prescribed a method for the simulation of interactions between

fluid and granular media by utilizing the augmented fluid and granular media simulation methods we

created.

We have only touched the begining of this area, as there are many ways for which this work could

be improved. In particular stability of the granular simulation still is an issue, as is the fact that we

must define a cutoff for which cells it is reasonable to label as cohesive. Furthermore the phenomology

of interacting continuum materials like sand and air to create sand dunes is a poorly explored area. We

hope to see further work published on novel physical phenomena, from both ourselves and others in the

future.

Appendix A

Calculus of Differential Forms

Differential forms provide a graded algebra of the family of differential operators. The different grades

of this algebra provide represent different types of fields suchas scalar fields, vector fields, tensor fields,

fluxes, and metrics through a unified representation. Through operators like the Hodge star and exterior

derivative, which operate on differential forms, we can reproduce many of the standard operators from

vector calculus.

We will begin with defining differential forms in a familiar context before we provide a more rigorous

definition. After that we will define some standard operators on differential forms and finally discuss

the tools necessary to do integration, which is necessary for the discrete setting.

A.1 Ordinary Integration

Let us begin this discussion by first thinking about the components of a standard integral. From a

traditional standpoint we are integrating over a subset of Rn like r0, 1sn

and will see a term like

ż

r0,1snfpx1, . . . , xnqdx1dx2 ¨ ¨ ¨ dxn

where the dxi are symbolic quantities that direct us of the axes for which we must integrate. For a more

general manifold Ω this sort of coordinate-based representation might not feasable. What we want is a

coordinate-free representation. One such representation is called differential forms. In this context we

consider the dxi as differential one-forms.

A.2 Tangent Bundles on Manifolds

Before we dig into the details of what differential forms are, let us first remember some elementary

details about manifolds and their tangent bundles. We do this because we are now trying to integrate on

manifolds which are a a sort of generalization of Euclidean space: a manifold is a collection of patches

that are locally Euclidian, so when we need to do local analysis we can transport ourselves to a Euclidian

space. We need the tangent bundle because that’s where the things that we want to integrate live.

Consider what a vector v is in an n-dimensional space like Rn, disregarding any manifold business

for now. It’s not a quantity in the space, but rather a quantity of a particular point. v sticks out from

43

Appendix A. Calculus of Differential Forms 44

a point p in a direction d, which means that it is really a equivalent to a tuple pp, dq. Returning to

manifolds, call the space of point/direction tuples the tangent bundle, or TM for a manifold M . For

each point p P M we have a linear space of tangent vectors denoted TpM . A vector at a point p is an

element of the tangent space TpM . The space of vector fields over M is the space of sections over TM ,

that is, the family of functions V : M Ñ TM such that for any x PM,V pxq P TxM .

As we have said earlier, manifolds are objects that are locally Euclidean. That means that we can

pick one coordinate chart φ : Rn ÑM for a neighborhood on the manifold M to correspond between that

neighborhood of M and a normal Euclidean space. For some orthogonal basis teiui“1¨¨¨n we can obtain

a basis for vectors in that neighborhood t BBxj uj“1¨¨¨n by evaluating the tangent map Dφ : TRn Ñ TM ,

which is defined by

Dφvpxq “d

dtfpx` tvq|t“0 “

ÿ

j

Bφ

Bxjpxqvj

at every point using the basis vectors.

The definition of a manifold provides a gluing map between coordinate charts taht allows for us to

write any vector Xp P TpM as

Xp “ÿ

j

Xj B

Bxj

p

This formulation of a vector on a manifold has a secondary notion that any vector is also a differential

operator that we can use on functions.

In a Euclidean space the tangent bundle almost leads us to the space of affine transformations

which discerns points and directions. The direction doesn’t need a designated point because there’s

an implicit translation of the vector at the origin to every other point in space. On a manifold that

implicit translation operator isn’t available and we have to build it using an affine connection like the

Levi-Cevita connection to obtain parallel transport, which is a sort of translation of vectors at one point

to vectors at another. With the connection we can compare vectors in TpM with vectors in TqM , so in

a Euclidian space we can endow the structure of an affine space by applying a Levi-Cevita connection

and quotienting the the space TM by vectors that can be parallel transported to one another.

A.3 Differential Forms

Now consider the space of linear functionals that take elements from TpM and map them to R. This

is called the dual space of TpM and is usually denoted by T˚pM and called the co-tangent space. The

collection of all co-tangent spaces is the co-tangent bundle T˚M .

The natural pairing between vectors v and dual vectors v is denoted by

xv, vy.

The space of differential zero-forms is on a manifold is simply the space of scalar functions f on the

manifold. When one takes the exterior differential operator d on a f one obtains a dual vector field.

That is, for each point p PM,x P TpM , dfxppq “ Dfppxq evaluates to a scalar value P R. This is precisely

a section of the co-tangent bundle, and so differential one-forms are identified with T˚M . Just like we

had the basis t BBdxi u for vectors, we can obtain a basis for the dual vectors by identifying the vector dx1

with the basis vector such that x BBxi , diy “ δij where δij is the Kronecker delta function. This produces


the family of differential one-forms

Ω1pMq “ tω “ÿ

i

ωidxiu.

We can then read the differentiation of a zero-form f as

df “ÿ

Bjdxj .

By using the wedge product we can extend the one-forms dx1, dx2 ¨ ¨ ¨ dxn to obtain differential two-

forms

Ω2pMq “ tω “ÿ

i

ÿ

j

ωijdxi ^ dxju.

By continuing this process one can obtain differential forms all the way up to ΩnpMq, which is a one

dimensional space comprised of scalar multiples of the determinant.

It’s worth noting that dimΩkpMq “ dimΩn´kpMq .

Intuitively, differential forms correspond to some common quantities. Though we won’t elucidate

the explanations too much quite yet, here are some some of the common quantities associated with the

forms in R3

Ω0pR3q ô Scalar functions

Ω1pR3q ô Vector functions or velocity fields

Ω2pR3q ô Vector functions or Flux

Ω3pR3q ô Scalar functions or Volume

A.4 Differentiating Differential Forms

The exterior derivative d allows for one to jump from differential k-forms to differential k+1-forms

through

dpfdx1 ^ dx2 ¨ ¨ ¨ ^ dxkq “ df ^ dx1 ^ dx2 ¨ ¨ ¨ ^ dxk.

For a finite dimensional space one can draw a diagram

Ω0 Ñd0 Ω1 Ñd1 Ω2 Ñd2 ¨ ¨ ¨Ωn

which is usually called the deRham complex, which is a graded algebra. In the above diagram we have

used di to denote the particular exterior derivatives from Ωi Ñ Ωi`1 and so far we are only able to

differentiate to the right.

The exterior derivative corresponds to some common quantities in when M is R3:

∇f “ d0f

∇ˆ u “ d1u

∇ ¨ u “ d2u

where f is a scalar function (identified with a 0-form) and u is a vector field (identified with a 1-form


and 2-form at the appropriate times).

A.5 The Hodge Star

As we have just implicitly mentioned, 1-forms and 2-forms can both correspond to vector fields, which

is a particular feature of being in R3. As we listed even earlier, there is a connection between 0-forms

and 3-forms in R3 as well, all thanks to the Hodge star, which has the prototype

‹k : ΩkpMq Ñ Ωn´kpMq.

The Hodge star is a bijective map that takes advantage of duality between differential forms of degree

k and n´ k in an n dimensional manifold. For a k-form ω we define ‹ω as the unique n´ k form such

that

ω ^ ‹ω “ dx1 ^ dx2 ¨ ¨ ¨ ^ dxn “ det

where det is the volume form defined by our metric. This operator defines an inner product for differential

forms defined by

xα, βy “ ‹pα^‹βq

which is easily confirmed to be an inner product (linearity by all operators being linear, positive-

definiteness by α^‹α “ 1).

This pairing is bijective, for as we noted earlier the space of k-forms and (n-k)-forms are the same

size,`

nn´k

˘

.

With the Hodge star operator we can now differentiate to the left as well through the following

commutative diagram:

Ωk Ωk`1

Ωn´k Ωn´pk`1q

‹k

dk

dn´k´1

δn´k

‹k`1

δk`1

where the bijectivity of the Hodge star allows for us to define the co-differential operator

δk`1 “ ‹´1k dn´k´1‹k`1 “ ‹n´kdn´k´1 ‹k`1 .

With the codifferential operator we can define more interesting operators Laplace-de Rham operator,

which is defined by δd` dδ.

A.6 Closed and Exact Forms

Because we now have a few differential operators we can define what closed, exact, co-closed, and co-

exact forms are. A k-form α is closed if dα “ 0 and co-closed if δα “ 0. α is exact if α “ dβ for some

pk ´ 1q-form β and co-exact if α “ δξ for some pk ` 1q-form ξ.


A key fact to note is that every exact form is closed and every co-exact form is co-closed by the fact

that dd “ 0 and δδ “ 0.

A.6.1 Hodge-Helmholtz Decomposition

A useful tool from this calculus is the Hodge-Helmholtz decomposition, which decomposes any k-form

as the sum of three forms: an exact form, a co-exact form, and a harmonic form.

α “ dβ ` δη ` γ

This decomposition is unique and guaranteed to exist for any C2

If we assume that our velocity field is a C2 1-form u this decomposition gives us the following:

u “ dp` δω ` γ.

where p is a 0-form representing pressure, ω is a 2-form representing vorticity, and γ, which is a

harmonic form such that ∆γ “ 0. The flows represented by harmonic forms are the space of Laminar

flows and have little relevance to the systems we will be looking at, so we will neglect it.

A.7 Integrating Differential Forms

Before we can uncover how to integrate we need to discuss what we are integrating over. Rather than

always integrating over the whole manifold, we will integrate over quantities called chains. It’s through

the discretization of chains into chain complexes that we obtain our numerical methods.

A.7.1 Cube Complexes

Before we define what chains are, lets first talk about cubes and cube complexes. Though the usual

discussion on this topic utilizes simplices and simplicial complexes, we choose to use cube complexes

because we use cube meshes in our discretization, not simplicial meshes. In this context a k-cube on

a manifold M is a chart from r0, 1s Ñ M , which we will usually identify with its image on M . When

M “ R3 a point is a 0-cube, a line is a 1-cube, a square is a 2-cube, etc.

A k-cube complex is a collection of cubes S such that the following two statements are true:

• For any two cubes α, β P S, αX β is a cube in S of dimension lower than those of α and β.

• For any cube α P S, α Ă β for some k-cube β P S.

The discretization I chose to use was a staggered grid as initially described by [?]. The essense of this

grid format is that rather than store values on only the vertices of a grid or on the centers, we store

quantities on the vertices, cells, and edges. This storage of a n dimensional staggered grid is generated

by a highly regular cube complex spanning one larger cube r0, 1sn. If we set the number of ticks per axis

to be Ni, one i per axis, we get that each n-cube is a translation ofŚ

r0,∆xis, the product of n intervals

of width ∆xi “1Ni

. We will only talk about uniform cubes where all Ni “ N for some N . That makes

each top-level cube a translation of r0, 1N s

n.


A.7.2 Chains

A k-chain is a finite sum of k-cubes with integral coefficients. That means that if we have a couple of

k-cubes c1, c2, ¨ ¨ ¨ , cl P S we can generate a k-chain by attaching coefficients a1, a2, ¨ ¨ ¨ , al P Z to create

a k-chain

σk “ a1c1 ` a2c2 ` ¨ ¨ ¨ ` alcl.

We will identify individual k-cubes c with the unit k-chain 1c.

The boundary of each k-cube to its 2k (k-1)-cubes on its boundary, so we define the boundary

operator B to be the chain generated by the following:

Bck “ c1k´1 ` c2k´1 ` ¨ ¨ ¨ ` c

2kk´1

where ck is a k-cube and the cik´1 are (k-1)-cubes.

so from any k-chain we can generate a (k-1)-chain through

σk´1 “ Bσk “ a1Bc1 ` a2Bc2 ` ¨ ¨ ¨ ` alBcl.

This boundary operator defines yet another graded algebra, the chain complex, that goes the opposite

way of the deRham complex.

C0 ¨ ¨ ¨ ÐBn´1Cn´1 ÐBn´2

Cn´2 ÐBn Cn

where Cl Ă S is the set of l-cubes in a k-cube complex.

A.7.2.1 Laplace-de Rham Operator

In the following discussion we will depend heavily on the Laplace-de Rham operator, which is a gen-

eralization of standard Laplacian operator to work for differential forms. We’ll refer to the Laplace-de

Rham as the Laplacian where the situation is unambiguous. It is defined as

Lu “ pdδ` δdqu

where u is a k-form. Because for 0-forms ω and n-forms γ we have

δω “ 0

dγ “ 0

In our case we only need the Laplacian of a 0-form, so δu “ 0. This allows us to simplify the Laplacian

operators we’ll use as

Lω “ δdu

Lγ “ dδu


It’s fairly clear that L is self adjoint according to the Riemannian metric γ by the relationship between

d and δ and that the Hodge star is self adjoint:

xpdδ` δdqu, vyγ “ xdδu, vyγ ` xδdu, vyγ

“ xδu, δvyγ ` xdu, dvyγ

“ xu, dδvyγ ` xu, δdvyγ

“ xu, pdδ` δdqvyγ

We can see the special n-form and 0-form cases by simply following the left or right sides of the addition

above.

In numerical methods solving the Poisson problem (∆u “ p) quite a few methods depend on having

a matrix that is self adjoint with respect to the standard metric. In the special cases, however, the

Laplace-de Rham matrix is clearly not diagonal with respect to the standard metric:

pd‹ dT‹qT“ pdδq

T“ δT d “ ‹d‹ dT

We therefore call the above the strong Laplace-de Rham operator (or strong Laplacian) and call the

following diagonalization of the operator the weak Laplace-de Rham operator (or weak Laplacian):

L “ dT ‹ d

which is clearly self adjoint by the self adjoint-ness of the Hodge star.

A.7.3 Integration

Now that we have a defined differential forms and chains we can finally discuss how one integrates on

cube complexes. Integration is done through a pairing between k-chains and differential forms, most

easily explained through the constitutive k-cubtes. By abusing notation slightly and referring to k-cubes

initially as sets and then as maps from r0, 1sk ÑM we obtain that

xω, σy “

ż

σ

ω “

ż

r0,1skωpσpxqqJpxqdx

where σ P Ck, ω P Ωk, and Jpxq “ detpDσpxqq is the Jacobian of σ evaluated at x.

Under this pairing Stokes theorem becomes the statement that the exterior derivative and boundary

operators are each others adjoints:

ż

σ

dω “ xdω, σy “ xω, Bσy “

ż

Bσ

ω.

A.8 Discrete Calculus on Differential Forms

In the previous section we discussed differential forms, but haven’t answered why this is relevant to our

problem of numerical simulation. The primary reason for all of this is that we really only have two

operators to compute in the above system, so by defining discrete analogues to those two operators on


a discrete space should be sufficient. The main question on how to define those two operators is the

same as in any problem: where do we store the discrete samples to approximate our space. We choose

to follow the practice of DEC (Discrete Exterior Calculus) [HNC08] which is to store objects where they

integrally make sense. By that we mean that a k-form is integrated over a k-dimensional manifold so

we store k-forms by their evaluated values on k-dimensional manifolds. For instance, if we were using

a simplicial complex (say a triangular mesh or a tetrahedreal mesh) 0-forms are stored on points and

2-forms are stored on trianges.

A.8.1 Exterior Derivative and Boundary Operators

This discretization makes generating the exterior derivative rather simple: assume we have k-polyhedron

Mk and a differential form σ stored as integrated quantities on the faces Mk, Mk´1i the integrated of

dσ over Mk. That is, we have some integrated values of σ for each face Mk´1 discrete values stored on

the faces σi defined by

σi “

ż

Mk´1i

σ

and by applying Stoke’s theorem we have

xdσ “

ż

Mk

dσ

“ÿ

i

sgni

ż

Mk´1i

σ

“ÿ

i

sgniσi.

where sgni denotes the sign of face Mk´1i with respect to Mk: `1 if they share orientation and ´1 if

not.

This leads to a simple definition of the exterior derivative of a complex of polyhedra: the discrete

exterior derivative of every polyhedron is the signed sum of its boundary elements depending on ori-

entation. This is the transpose of the signed boundary operator, which is a natural result from the

adjointness of the exterior derivative and the boundary, and so this operator is generally very easy to

compute.

The other main component to consider is how to define the Hodge Star in the discrete case.

A.8.2 Hodge Star

In our case we are using a cube complex which means that we are storing quantities on k-cubes. This

turns out to be identical to what is commonly used as staggered grid methods, which we discuss in the

main body of this thesis.

A.9 Pressure Projection

TODO: Maybe i’ll move pressure projection differential forms stuff to here?

Appendix B

Discrete Differential Forms on

Cubical Complexes

Appendix A provides the preliminaries of doing exterior calculus on cubical complexes, and here we will

discuss the explicit construction of those operators.

We will only consider a particular family of cubical complexes: those that only have vertices from Z3

and are axis-aligned. This simplification is what will allow us to tie our work with standard staggered-

grid methods. Because of our use of a regular cubical grid, there are several convenient simplifications

that can be made in implementation. Some of these will be placed in this section while others will be

placed in Appendix B.

B.1 Level-set based Hodge Star

Let us begin with the derivations of the basic exerior calculus operators on cubical complexes, which

will be used in our discussion of a levelset hodge-star: the Boundary operator and the Hodge star.

Because of the structure of our chosen family of cubical complexes there is a very convenient indexing

scheme that we will use for the rest of this article: If in Rn we let a pi, j, kq P Z3 denote the index of

the unit n-cube that is centered at that location, we can uniquely index every k-cube by the location

of center. The set of centers is isomorphic to pZ2qn. In fact the centers of all of the k-cubes can be

represented by the subset of pZ2qn such that k of the numbers in the n-tuple are integral (i.e p2, 3, 4.5q

represents a 2-form and p0, 0, 0q represents a 3-form). From herein we will denote a n-cube by its index.

B.1.1 Connection to Staggered Grid Methods

In a staggered grid configuration one usually maintains a couple of grid types: one for vertices, one for

cells, and n grids for fluxes along each coordinate axis.

The cell grid is usually used to represent pressure, which is a volumetric quantity and therefore

representable as a n-form. The flux grids are used to represent one coordinate of the velocity at a time

and are exactly where our n´ 1 forms are placed — and a common interpretation of n´ 1 forms is flux.

The forms of lower dimension are usually not used.

51

Appendix B. Discrete Differential Forms on Cubical Complexes 52

B.1.2 Boundary Operator

Because of the regularity of our cubical complexes (the vertices form a lattice), Each n-cube is surrounded

by 2˚n edges, two per coordinate axis. Per axis, one of the boundary n´1 cubes has a lower index in the

grid and one has a higher index in that chosen axis. We choose to give the lower index a different sign

and the higher index the same sign. For a cube pi, j, kq this makes the relevant entries in the boundary

map the following:

Bpi,j,kq,pi´ 12 ,j,kq

“ ´1

Bpi,j,kq,pi` 12 ,j,kq

“ 1

Bpi,j,kq,pi,j´ 12 ,kq

“ ´1

Bpi,j,kq,pi,j` 12 ,kq

“ 1

Bpi,j,kq,pi,j,k´ 12 q

“ ´1

Bpi,j,kq,pi,j,k` 12 q

“ 1.

In implementation this is usually performed on a 3-dimensional array of cubes forming a filled rect-

angular prism. If we set the height, width, and depth of this prism to by M,N,O respectively we obtain

a boundary operator matrix with dimensions NMO by pN ` 1qMO `NpM ` 1qO `NMpO ` 1q.

B.1.3 Hodge Star

First let us go over how the Hodge Star on uniform grids is the same as standard finite difference

operators.

B.1.3.1 Hodge Star on Uniform Grids

On a staggered grid with uniform spacing the diagonal Hodge Star turns out to be scalar. This begins

with remembering that the diagonal Hodge Star is defined as the ratio of the volumes of the cube and

then noting that the volume of a k-cube is ∆xk, so the diagonal Hodge Star on is

‹k “∆xn´k

∆xk.

‹n “1

∆xn

‹n´1 “∆x1

∆xn´1.

Appendix B. Discrete Differential Forms on Cubical Complexes 53

B.1.3.2 Differential and Codifferential Operator

Now that we have the boundary and the differential operator are adjoint, we see that the discrete

representation of the differential operator is

d “ B˚.

Then if we remind ourselves that the codifferential operator is defined by δn “ p‹n´1q´1pdnq

˚‹n we

quickly see that

δn “ p‹n´1q´1pdnq

˚‹n “

∆xn´1

∆xpdnq

˚ 1

∆xn“

∆xn´1

∆x

1

∆xnpdnq

˚“

1

∆x2pdnq

˚

so the codifferential operator is simply the boundary operator scaled by 1∆x2 .

B.1.3.3 Constructing the Laplace-deRham

For n-forms the Laplace-deRham operator is the standard finite difference Laplacian, with a 5-point

stencil in 2 dimensions and 7-point stencil in 3 dimensions comprising of ´ 1∆x2 off diagonal bands and

positive entries to main diagonal to make each row sum zero. To see this consider the following:

First the explicit representation of Laplace-deRham is

dδ “1

∆x2ddT “

1

∆x2BT B.

To compute the value for the row representing α “ pi, j, kq and column representing α “ pi, j, kq recall

from linear algebra that the entry for α, α is Bα,: ¨ Bα,: where Ba,: is vector of the a column in B. With

that information we can directly compute the entries of the matrix from three cases:

• α “ α

Since all entries line up and all of the entries are ˘1 we get 2 ˚n nonzero entries: one for each face

on the boundary of the cube. Therefore pdδqα,α “1

∆x2 2n

• α shares a face with α

Here the only shared nonzero entries are those representing the shared face. This face, by con-

struction, is the upper face of one and the lower face of another, so the signs of the only nonzero

entries are different. Therefore pdδqα,α “ ´1

∆x2

• α shares no faces with α.

Since no nonzero entries coincide the dot product must result in 0.

Appendix C

Numerical Optimization

In order to implement the methods described above it was necessary to implement and utilize a Quadratic

Programming solver in two separate locations, with one of the systems actually being a Linear Com-

plementary Problem. Because of the structure of systems being solved, it seemed prudent to use one

particular solver called Modified Proportions with Reduced Gradient Projections (MPRGP) and there

was need to modify the solver to make it usable in our systems. Some of these modifications were utilized

in [NGCL09] but not detailed in the paper, but we will discuss them fully in the technical details section.

Therefore, we thought it prudent to provide some introduction to LCP and QP problems and solvers.

First we will discuss some basic notations of constrained optimization. We will then overview the

formulations of LCP/QP problems. Finally, we will discuss the duality between LCP and QP problems,

which allows for us to use one solver for two types of systems.

C.0.4 Constrained Optimization

Constrained optimization is the task of minimizing some function f : Rn ÞÑ R under equality and

inequality constraints. Let us notate these constraints all as ci where i P E are equality constraints and

i P I are inequality constraints.

The feasible set, the set of legitimate values for x in this constrained space, is defined as

Ω “ tx|cipxq “ 0, i P E , cipxq ě 0, i P Iu

The boundary of Ω is quite important so there’s a fair bit of terminology around it. The active set Apxqis the set of constraints for which cipxq “ 0 and Fpxq, the inactive set, denotes the complement of Apxq.The set of constraints that are active becomes a fundamental tool in the algorithms to come.

The Karush-Kuhn-Tucker (KKT) conditions are first order necessary conditions for a local minimum

for an optimization problem with equality and inequality constraints and can be seen as a generalization

54

Appendix C. Numerical Optimization 55

of Lagrange multipliers to support inequality constraints. They can be seen as the system

∇xLpx, λq “ 0

cipxq “ 0, @i P E

cipxq ě 0, @i P I

λi ě 0, @i P I

λicipxq ě 0, @i P E Y I

where ∇x is the gradient with respect to x and

Lpx, λq “ fpxq ´ÿ

iPEYIλicipxq.

C.0.5 Linear Complementary Problems

In my exploration of LCP solvers two distinct formulations of the problem definition that were used to

develop algorithms. The standard formulation of linear complimentary problems is given by

w “ Mz ` q (C.1)

w ě 0 (C.2)

z ě 0 (C.3)

wT z “ 0 (C.4)

where M usually assumed to be symmetric positive definite. The SPD assumption exists because so-

lutions are only known if it is SPD. The reason for this assumption can be understood through the

connection between LCP and quadratic programming. Any LCP instance can be seen a the solution to

the Karush-Kunn-Tucker (KKT) constraints of the quadratic problem instance

minuφpzq “

1

2zTMz ` qT z (C.5)

given inequality constraints that zi ě 0 for all i. If we allow for equality constraints we obtain the mixed

LCP (mLCP) formulation of

Au` Cv ` a “ 0 (C.6)

CTu`Bv ` b “ w (C.7)

vTw ě 0 (C.8)

v, w ě 0 (C.9)

(C.10)

which solves a slightly different quadratic system:

minzφpzq “

1

2zTQz ` rT z (C.11)

Appendix C. Numerical Optimization 56

where v ě 0 and

Q “

«

A C

CT B

ff

z “

«

u

v

ff

, r “

«

a

b

ff

.

The equivalence of the two formulations is rather easy to formulate. Given an LCP pM, qq we can

create an equivalent mLCP by setting pA,B,C, a, bq to be p0,M, 0, 0, qq.

An mLCP can be reduced to an LCP by rearranging terms in the mLCP formulation by solving for

u and substituting,which provides us with:

M “ B ´ CTA´1C (C.12)

q “ “ b´ CTA´1a (C.13)

.

C.0.6 LCP and Karush-Kuhn-Tucker Conditions

Though I’ve mentioned that LCPs can be written as the solutions to a quadratic programming instance,

it can also be shown that the KKT first order conditions for a quadratic programming instance can

easily be described in terms of an LCP. The natural instance of a quadratic programming problem is a

quadratic functional bounded by linear inequality constraints like this:

minx

1

2xTGx` xT c (C.14)

Ax ě b (C.15)

(C.16)

Under the standard LCP formulation the only bound on x is positive and other constraints are not

obviously represented. By applying the KKT conditions on this system, we see that a solution to the

above quadratic programming instance must be a solution to the LCP given by the system below where

a slack variable λ must be introduced:

«

G ÁT

A 0

ff«

x

λ

ff

“

«

ć

b

ff

.

Given that if the above system has full rank and G is positive symmetric definite the solution is unique

and so solving the LCP is equivalent to solving the quadratic problem. Therefore we see that LCP is

equivalent to QP.

C.0.7 Complimentary and Minimization Duality

Maintaining the complimentary condition in an iterative process is quite difficult. When the constraint

variables are strictly positive instead of directly forcing the complimentary condition in iterative algo-

rithms, it’s simpler to treat the constraint as a minimization problem. That is, two vectors u, v ě 0 are

complimentary to each other if and only if for any index i, minpui, viq “ 0 so we can reformulate our com-

plimentary equality constraint as a minimization constraint on the min of the two vectors coefficient-wise.

This will be used to determine convergence tests in the iterative algorithms that follow.

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Documents

by Michael Tao - Dynamic Graphics Projectmtao/files/publications/msc.pdf · to the existing technologies in graphics for simulating granular materials, we choose a continuum based